# Drawing conclusions from impulse response of a discrete LTI system

I have an impulse response of an LTI system which is

h(n)= n*u(n)-u(n-2) where n=[0,3] (1)


Now regarding the stability and causality of this system, i've drawn the conclusion that the system is causal since the output only relies on past or present values of n and it is stable since n is bounded. In this case n=0,1,2,3. Correct me here if i'm wrong but in case h(n) was u(n) - u(n-2) without knowing anything about n, it would stil be stable right? And if h(n) is the same as the original (1) but again without knowing anything about n it would be unstable correct?

Now i want to analyze this h(n) in odd and even components and also find the linear difference equation of h(n) ,but i can't find anything in my textbook that is of any help. It's the first time i've attended a signals and systems course and i'm really struggling to understand a few things, so any help here would be greatly appreciated.

• Is $h[n]$ given by $n\cdot (u[n]-u[n-2])$ or by $n\cdot u[n] - u[n-2]$? – Matt L. Jul 7 at 16:06
• @MattL. second one – Notoriousphd Jul 7 at 16:22

1. Your textbook should state somewhere the definition of even and odd parts of a (real-valued) sequence: $$h_e[n]=\frac12\big(h[n]+h[-n]\big)\\h_o[n]=\frac12\big(h[n]-h[-n]\big)$$
2. In this case (FIR filter), the difference equation is simply given by the convolution of the input sequence $$x[n]$$ with the impulse response $$h[n]$$: $$y[n]=\sum_kx[k]h[n-k]$$